Contents

Mathematics:Real NumbersFunctions and Inverse FunctionsComposite FunctionsInverse Composite FunctionsSome Worked Problems on IndicesFormulae for regular polygonsSome similar solidsSurd form of trigonometrical ratios

Rational numbers are numbers that can be expressed as the quotient of 2 integers a and b provided b doesn't = 0. All integers are rational as all are divisible by 1. Repeating and terminating decimals are rational, e.g. 1.3333... and 0.123. The set of rational numbers is denoted Q.

Surds, or irrational numbers, are non-repeating infinite numbers such as pi. pi is the ratio of the circumference of a circle to its diameter in radians. 3.1416 is a rational approximation to 4 decimal places. Other examples of surds include root3, pi/4, 4 - 2 root11. The set of surds is denoted by J.

Real numbers, R, are all rational or irrational numbers. (They exclude complex numbers such as root-1, denoted by i).

Complex numbers, denoted by z, are numbers in the form a + ib, where i = root -1 and a and b are real numbers. a is the real part, Rz, and b, the imaginary part Iz. e.g. z = 3 + 4i, where 3 = Rz, 4 = Iz.

Functions and Inverse Funtions:

If a given value of x has only 1 corresponding y value, y is a function of x, e.g. y = 2x + 7, and to denote the function of x, f(x) = 2x + 7. x values are referred to as the domain and y values are referred to as the range. Each pair of corresponding values for x and y constitute an ordered pair. An equation such as y = x2 does not constitute a function as x can have a ± value, eg. y = 4, x = ±2, and hence, more than one possible value. For y = x3 is a function, as for any value of x, there is only one possible value for y.

To find the inverse of a function, simply interchange the range values with the domain values for x and y. The inverse of f(x) is denoted by f -1(x).e.g. f(x) = x + 4/ 5, the inverse function is found by reversing the order of operations:f -1(x) = 5x - 4. The inverse of a function can help in the solution of equations.

Volume and area - similar solids:The surface area of similar solids are proportional to the squares of their linear dimensions. The volumes of similar solids are proportional to the cubes of their linear dimensions.

2 cylinders are similar:Height of larger / height of smaller = Diameter of larger / diameter of smallerSurface area of larger / surface area of smaller = (height of larger)2 / (height of smaller)2 = (Diameter of larger)2 / (diameter of smaller)2Volume of larger / volume of smaller = (Height of larger)3 / (height of smaller)3 = (Diameter of larger)3 / (diameter of smaller)32 spheres are similar:Surface area of smaller / surface area of larger = (Diameter of smaller)2 / (diameter of larger)2Surface area of the smaller sphere = area of larger sphere / denominator.Volume of smaller / volume of larger = (Diameter of smaller)3 / (diameter of larger)3Volume of smaller sphere = Volume of larger / denominator.